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CS344: Introduction to Artificial Intelligence (associated lab: CS386). Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture–4: Fuzzy Control of Inverted Pendulum + Propositional Calculus based puzzles. Lukasiewitz formula for Fuzzy Implication. - PowerPoint PPT Presentation
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CS344: Introduction to Artificial Intelligence
(associated lab: CS386)
Pushpak BhattacharyyaCSE Dept., IIT Bombay
Lecture–4: Fuzzy Control of Inverted Pendulum + Propositional Calculus based
puzzles
Lukasiewitz formulafor Fuzzy Implication
t(P) = truth value of a proposition/predicate. In fuzzy logic t(P) = [0,1]
t( ) = min[1,1 -t(P)+t(Q)]QP
Lukasiewitz definition of implication
Use Lukasiewitz definition t(pq) = min[1,1 -t(p)+t(q)] We have t(p->q)=c, i.e., min[1,1 -t(p)+t(q)]=c Case 1: c=1 gives 1 -t(p)+t(q)>=1, i.e., t(q)>=a Otherwise, 1 -t(p)+t(q)=c, i.e., t(q)>=c+a-1 Combining, t(q)=max(0,a+c-1) This is the amount of truth transferred over the
channel pq
Fuzzification and Defuzzification
Precise number(Input)
Fuzzy Rule Precise number
(action/output)Fuzzificatio
nDefuzzification
Eg: If Pressure is high AND Volume is low then make Temperature Low
))(),(min()( QtPtQPt
Pressure/Volume/Temp
High Pressure
ANDING of Clauses on the LHS of implication
Low Volume
Low Temperature
P0 V0T0Mu(P0)<Mu(V0)
Hence Mu(T0)=Mu(P0)
Fuzzy InferencingCore
The Lukasiewitz rulet( ) = min[1,1 + t(P) – t(Q)]An example
Controlling an inverted pendulum
QP
θ dtd /.
= angular velocity
Motor i=current
The goal: To keep the pendulum in vertical position (θ=0)in dynamic equilibrium. Whenever the pendulum departs from vertical, a torque is produced by sending a current ‘i’
Controlling factors for appropriate current
Angle θ, Angular velocity θ.
Some intuitive rules
If θ is +ve small and θ. is –ve small
then current is zero
If θ is +ve small and θ. is +ve small
then current is –ve medium
-ve med
-ve small
Zero
+ve small
+ve med
-ve med
-ve small Zero +ve
small+ve med
+ve med
+ve small
-ve small
-ve med
-ve small
+ve small
Zero
Zero
Zero
Region of interest
Control Matrix
θ.
θ
Each cell is a rule of the form
If θ is <> and θ. is <>
then i is <>
4 “Centre rules”
1. if θ = = Zero and θ. = = Zero then i = Zero
2. if θ is +ve small and θ. = = Zero then i is –ve small
3. if θ is –ve small and θ.= = Zero then i is +ve small
4. if θ = = Zero and θ. is
+ve small then i is –ve small
5. if θ = = Zero and θ. is –ve small then i is +ve small
Linguistic variables
1. Zero
2. +ve small
3. -ve small
Profiles
-ε +ε ε2-ε2
-ε3 ε3
+ve small-ve small
1
Quantity (θ, θ., i)
zero
Inference procedure
1. Read actual numerical values of θ and θ.
2. Get the corresponding μ values μZero, μ(+ve small), μ(-
ve small). This is called FUZZIFICATION3. For different rules, get the fuzzy i values from
the R.H.S of the rules.4. “Collate” by some method and get ONE current
value. This is called DEFUZZIFICATION5. Result is one numerical value of i.
if θ is Zero and dθ/dt is Zero then i is Zeroif θ is Zero and dθ/dt is +ve small then i is –ve smallif θ is +ve small and dθ/dt is Zero then i is –ve smallif θ +ve small and dθ/dt is +ve small then i is -ve medium
-ε +ε ε2-ε2
-ε3 ε3
+ve small-ve small
1
Quantity (θ, θ., i)
zero
Rules Involved
Suppose θ is 1 radian and dθ/dt is 1 rad/secμzero(θ =1)=0.8 (say)μ +ve-small(θ =1)=0.4 (say)μzero(dθ/dt =1)=0.3 (say)μ+ve-small(dθ/dt =1)=0.7 (say)
-ε +ε ε2-ε2
-ε3 ε3
+ve small-ve small
1
Quantity (θ, θ., i)
zero
Fuzzification
1rad
1 rad/sec
Suppose θ is 1 radian and dθ/dt is 1 rad/secμzero(θ =1)=0.8 (say)μ +ve-small(θ =1)=0.4 (say)μzero(dθ/dt =1)=0.3 (say)μ+ve-small(dθ/dt =1)=0.7 (say)
Fuzzification
if θ is Zero and dθ/dt is Zero then i is Zeromin(0.8, 0.3)=0.3
hence μzero(i)=0.3if θ is Zero and dθ/dt is +ve small then i is –ve small
min(0.8, 0.7)=0.7hence μ-ve-small(i)=0.7
if θ is +ve small and dθ/dt is Zero then i is –ve smallmin(0.4, 0.3)=0.3
hence μ-ve-small(i)=0.3if θ +ve small and dθ/dt is +ve small then i is -ve medium
min(0.4, 0.7)=0.4hence μ-ve-medium(i)=0.4
-ε +ε-ε2
-ε3
-ve small
1zero
Finding i
0.4
0.3Possible candidates:
i=0.5 and -0.5 from the “zero” profile and μ=0.3i=-0.1 and -2.5 from the “-ve-small” profile and μ=0.3i=-1.7 and -4.1 from the “-ve-small” profile and μ=0.3
-4.1-2.5
-ve small-ve medium
0.7
-ε +ε
-ve smallzero
Defuzzification: Finding iby the centroid method
Possible candidates:i is the x-coord of the centroid of the areas given by the
blue trapezium, the green trapeziums and the black trapezium
-4.1-2.5
-ve medium
Required i valueCentroid of three trapezoids
Propositional Calculus and Puzzles
Propositions
− Stand for facts/assertions− Declarative statements
− As opposed to interrogative statements (questions) or imperative statements (request, order)
Operators
=> and ¬ form a minimal set (can express other operations)- Prove it.
Tautologies are formulae whose truth value is always T, whatever the assignment is
)((~),),(),( NIMPLICATIONOTORAND
Model
In propositional calculus any formula with n propositions has 2n models (assignments)
- Tautologies evaluate to T in all models.
Examples: 1)
2)
- e Morgan with AND
PP
)()( QPQP
Semantic Tree/Tableau method of proving tautology
Start with the negation of the formula
α-formula
β-formula α-formula
pq
¬q¬ p
- α - formula
- β - formula
)]()([ QPQP
)( QP
)( QP
- α - formula
Example 2:
B C B CContradictions in all paths
X
α-formula ¬A ¬C
¬A ¬B ¬A ¬B
AB∨C
AB∨C A
B∨CA
B∨C
(α - formulae)
(β - formulae)
(α - formula)
)]()()([ CABACBA
)( CBA
))()(( CABA
)( BA
))( CA
A puzzle(Zohar Manna, Mathematical Theory of Computation, 1974)
From Propositional Calculus
Tourist in a country of truth-sayers and liers Facts and Rules: In a certain country,
people either always speak the truth or always lie. A tourist T comes to a junction in the country and finds an inhabitant S of the country standing there. One of the roads at the junction leads to the capital of the country and the other does not. S can be asked only yes/no questions.
Question: What single yes/no question can T ask of S, so that the direction of the capital is revealed?
Diagrammatic representation
S (either always says the truthOr always lies)
T (tourist)
Capital
Deciding the Propositions: a very difficult step- needs human intelligence P: Left road leads to capital Q: S always speaks the truth
Meta Question: What question should the tourist ask The form of the question Very difficult: needs human
intelligence The tourist should ask
Is R true? The answer is “yes” if and only if
the left road leads to the capital The structure of R to be found as
a function of P and Q
A more mechanical part: use of truth table
P Q S’s Answer
R
T T Yes T
T F Yes F
F T No F
F F No T
Get form of R: quite mechanical From the truth table
R is of the form (P x-nor Q) or (P ≡ Q)
Get R in English/Hindi/Hebrew… Natural Language Generation: non-
trivial The question the tourist will ask is
Is it true that the left road leads to the capital if and only if you speak the truth?
Exercise: A more well known form of this question asked by the tourist uses the X-OR operator instead of the X-Nor. What changes do you have to incorporate to the solution, to get that answer?